# Applying Newton's 2nd to a rocket

#### dwsmith

##### Well-known member
A rockt with mass $$M$$ is sitting motionless in space. The rocket is powered by steam with velocity $$V_s$$ and mass per time $$C = \frac{dm}{dt}$$. Write an equation for the rockets acceleration in terms of $$M, {} V_s$$, and $$C$$.

So $$\mathbf{F}_s = \mathbf{F}_r$$ by Newton's 3rd. By Newton's 2nd,
\begin{align}
\mathbf{F}_s &= \mathbf{F}_r\\
\frac{d(mV_s)}{dt} &= \frac{d(mV_r)}{dt}\\
m\frac{dV_s}{dt} + V_s\frac{dm}{dt} &= m\frac{dV_r}{dt} + V_r\frac{dm}{dt}\\
V_s\cdot C &= ma_r
\end{align}
Here we assumed the steam isn't accelerating and $$\frac{dm}{dt} = 0$$ on the RHS. I understand assuming the steam acceleration is zero, but why is the $$\frac{dm}{dt}$$ on the RHS zero and not $$C$$?